Table 1 Allele configurations and their expected frequencies.

From: Linkage disequilibrium under polysomic inheritance

Allele configuration

Freq. notation

Double non-identity

Expectation

Expectation notation

\(p_Aq_B\)

\(p_Aq_B(p_X + q_X)\)

\(p_Ap_Xq_Bq_X\)

Digenic

\(Z_{B \ldots }^{A \ldots }\)

\(P_1\)

 

1

  

\(E_1\)

\(Z_{ \cdot B \ldots }^{A \cdot \ldots }\)

\(P_2\)

 

1

  

\(E_2\)

\(Z_{ \cdot \ldots\!\! |B \ldots }^{A \ldots \!\!| \cdot \ldots }\)

\(P_3\)

 

1

  

\(E_3\)

Trigenic

\(Z_{B \cdot \ldots }^{AA \ldots },\,Z_{BB \ldots }^{A \cdot \ldots }\)

\(P_4\)

 

2

\( - P\)

 

\(E_4\)

\(Z_{ \cdot \ldots \!\! |B \ldots }^{AA \ldots \!\!| \cdot \ldots },\,Z_{BB \ldots \!\!| \cdot \ldots }^{ \cdot \ldots \!\!|A \ldots }\)

\(P_5\)

 

2

\( - P\)

 

\(E_5\)

\(Z_{B \ldots \!\! | \cdot \ldots }^{A \ldots \!\! |A \ldots },\,Z_{B \ldots \!\! |B \ldots }^{A \ldots \!\! | \cdot \ldots }\)

\(P_6\)

 

2

\( - {{{\mathrm{{\Pi}}}}}\)

 

\(E_6\)

\(Z_{ \cdot B \ldots \!\!| \cdot \ldots }^{A \cdot \ldots\!\! |A \ldots },Z_{ \cdot B \ldots \!\! |B \ldots }^{A \cdot \ldots \!\! | \cdot \ldots }\)

\(P_7\)

 

2

\( - {{{\mathrm{{\Pi}}}}}\)

 

\(E_7\)

\(Z_{ \cdot \ldots \!\!| \cdot \ldots \!\!|B \ldots }^{A \ldots \!\! \left| {A \ldots } \!\! \right| \cdot \ldots },Z_{ \cdot \ldots \!\! |B \ldots \!\! |B \ldots }^{A \ldots \!\! \left| { \cdot \ldots } \!\! \right| \cdot \ldots }\)

\(P_8\)

 

2

\( - {{{\mathrm{{\Pi}}}}}\)

 

\(E_8\)

\(Z_{ \cdot B \ldots }^{AA \cdot \ldots },Z_{ \cdot BB \ldots }^{A \cdot \cdot \ldots }\)

\(P_9\)

 

2

\( - P\)

 

\(E_9\)

Quadgenic

Dihaplotypic

\(Z_{BB \ldots }^{AA \ldots }\)

\(P_{10}\)

\({{\Theta }}_1\)

1

\(- {P}\)

\({{\Theta }}_1\)

\(E_{10}\)

\(Z_{B \ldots \!\! |B \ldots }^{A \ldots \!\! |A \ldots }\)

\(P_{11}\)

\({{\Theta }}_2\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Theta }}_2\)

\(E_{11}\)

Trihaplotypic

\(Z_{B \ldots \!\! | \cdot B \ldots }^{A \ldots \!\! |A \cdot \ldots }\)

\(P_{12}\)

\({{\Gamma }}_1\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Gamma }}_1\)

\(E_{12}\)

\(Z_{B \cdot \ldots \!\! |B \ldots }^{AA \ldots \!\! | \cdot \ldots },Z_{BB \ldots \!\! | \cdot \ldots }^{A \cdot \ldots \!\! |A \ldots }\)

\(P_{13}\)

\({{\Gamma }}_2\)

2

\(- {P} - {{{\mathrm{{\Pi}}}}}\)

\(2{{\Gamma }}_2\)

\(E_{13}\)

\(Z_{B \ldots \!\! | \cdot \ldots \!\! |B \ldots }^{A \ldots \!\! |A \ldots \!\! | \cdot \ldots }\)

\(P_{14}\)

\({{\Gamma }}_3\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Gamma }}_3\)

\(E_{14}\)

\(Z_{B \cdot B \ldots }^{AA \cdot \ldots }\)

\(P_{15}\)

\({{\Gamma }}_4\)

1

\(- {P}\)

\({{\Gamma }}_4\)

\(E_{15}\)

Quadhaplotypic

\(Z_{ \cdot B \ldots \!\!| \cdot B \ldots }^{A \cdot \ldots \!\! |A \cdot \ldots }\)

\(P_{16}\)

\({{\Delta }}_1\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Delta }}_1\)

\(E_{16}\)

\(Z_{ \cdot \cdot \ldots \!\! |BB \ldots }^{AA \ldots \!\! | \cdot \cdot \ldots }\)

\(P_{17}\)

\({{\Delta }}_2\)

1

\(- {P}\)

\({{\Delta }}_2\)

\(E_{17}\)

\(Z_{ \cdot B \ldots \!\! | \cdot \ldots \!\! |B \ldots }^{A \cdot \ldots \!\! |A \ldots \!\! | \cdot \ldots }\)

\(P_{18}\)

\({{\Delta }}_3\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Delta }}_3\)

\(E_{18}\)

\(Z_{ \cdot \cdot \ldots \!\! |B \ldots \!\! |B \ldots }^{AA \ldots \!\! | \cdot \ldots \!\! | \cdot \ldots },Z_{BB \ldots \!\! \left| { \cdot \ldots } \!\! \right| \cdot \ldots }^{ \cdot \cdot \ldots \!\! |A \ldots \!\! |A \ldots }\)

\(P_{19}\)

\({{\Delta }}_4\)

2

\(- {P} - {{{\mathrm{{\Pi}}}}}\)

\(2{{\Delta }}_4\)

\(E_{19}\)

\(Z_{ \cdot \ldots \!\! | \cdot \ldots \!\! |B \ldots \!\! |B \ldots }^{A \ldots \!\! |A \ldots \!\! | \cdot \ldots \!\! | \cdot \ldots }\)

\(P_{20}\)

\({{\Delta }}_5\)

1

\( - {{{\mathrm{{\Pi}}}}}\)

\({{\Delta }}_5\)

\(E_{20}\)

\(Z_{\cdot \cdot BB\ldots}^{AA \cdot \cdot \ldots}\)

\(P_{21}\)

\({{\Delta }}_6\)

1

\(- {P}\)

\({{\Delta }}_6\)

\(E_{21}\)

\(Z_{ \cdot \cdot B \ldots \!\! |B \ldots }^{AA \cdot \ldots \!\! | \cdot \ldots },Z_{BB \cdot \ldots \!\! | \cdot \ldots }^{ \cdot \cdot A \ldots \!\! |A \ldots }\)

\(P_{22}\)

\({{\Delta }}_7\)

2

\(- {P} - {{{\mathrm{{\Pi}}}}}\)

\(2{{\Delta }}_7\)

\(E_{22}\)

  1. Z denotes an allele configuration, P and \({{{\mathrm{{\Pi}}}}}\) denote the single non-identity coefficient within and between individuals. Superscripts and subscripts denote the genotype patterns at two target loci, respectively. The vertical bars separate individuals, · denotes an allele copy in any form, and\( \ldots \)denotes the remaining allele copies. The allele copies in the same position are in the same haplotype. For example, \(Z_{B \! \ldots }^{A \! \ldots }\), \(Z_{ \cdot B \! \ldots }^{A \cdot \! \ldots }\) and \(Z_{ \cdot \! \ldots |B \! \ldots }^{A \! \ldots | \cdot \! \ldots }\) denote the two target alleles are in the same haplotype, different haplotypes in the same individual, and different individuals, respectively.
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